Weyl groups (experimental features)

FPGroup(W::WeylGroup) -> FPGroup
fp_group(W::WeylGroup) -> FPGroup

Construct a group of type FPGroup that is isomorphic to W.

If one needs the isomorphism then isomorphism(::Type{FPGroup}, W::WeylGroup) can be used instead.

isomorphism(::Type{FPGroup}, W::WeylGroup) -> Map{WeylGroup, FPGroup}

Return an isomorphism from W to a group H of type FPGroup.

H will be the quotient of a free group with the same rank as W, where we have the natural 1-to-1 correspondence of generators, modulo the Coxeter relations of W.

Isomorphisms are cached in W, subsequent calls of isomorphism(FPGroup, W) yield identical results.

If only the image of such an isomorphism is needed, use fp_group(W).

PermGroup(W::WeylGroup) -> PermGroup
permutation_group(W::WeylGroup) -> PermGroup

Construct a group of type PermGroup that is isomorphic to W.

If one needs the isomorphism then isomorphism(::Type{PermGroup}, W::WeylGroup) can be used instead.

isomorphism(::Type{PermGroup}, W::WeylGroup) -> Map{WeylGroup, PermGroup}

Return an isomorphism from W to a group H of type PermGroup. An exception is thrown if no such isomorphism exists.

The generators of H are in the natural 1-1 correspondence with the generators of W.

If the type of W is irreducible and not $E_6$ or $E_7$, then the degree of H is optimal. See [Sau14] for the optimal permutation degrees of Weyl groups.

Isomorphisms are cached in W, subsequent calls of isomorphism(PermGroup, W) yield identical results.

If only the image of such an isomorphism is needed, use permutation_group(W).